I wonder if this is why I've seen metal buttons more often in hospital elevators than elsewhere.
That would be very expensive for very little benefit - five minutes is more than long enough for multiple people to use the same buttons at a busy hospital during the day.
It's much better to promote behaviors like washing hands before touching food or mucous membranes (like rubbing your eyes) and using alcohol-based hand sanitizers when that isn't practical. That's why you see hand sanitizer dispensers every few feet at (good) hospitals, and why infection rates plummeted when hospitals introduced them.
In addition, never touch those buttons with the tips of your fingers. Use the knuckle of your pinky instead - it's much less common to accidentally use that to touch a mucous membrane later on, so it's slightly better.
It's a matter of defense-in-depth. When the traffic is heavy enough to have less than 5 minutes between contacts, how many more people would be infected after a week?
As for cost, it would be more expensive but less so than you might think. With how thin the plating layer is, even for heavy wear, the material cost is under $500 per square meter. For small objects like buttons, door knobs, and hand rails it's negligible compared to installation and other costs. By switching to a cheaper base material it could even be cheaper over all.
Combining suggestions above, perhaps touching it with a ring on your finger would be even better (e.g. a silver ring).
Alcohol hand gels give a false sense of cleanliness though. Some hospitals have problems with Norovirus, and it's likely members of the public who've recently had vomiting think it's okay for them to visit a relative because of all the handgel.
It's not perfect, but it's empirically better than the previous situation.